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--- forcebalance:equation_of_motion [2023/11/21 14:06] – comment cjoens
+++ forcebalance:equation_of_motion [2023/11/22 12:42] (current) – [Current Density Operator] cjoens
@@ Line 16: / Line 16: @@
   \quad .
 \end{equation}
-Under the following conditions, the EoM for the electron current $J(x,t)$ becomes
+=====Fermionic Field Operator=====
+The EoM for the electronic creation and annihilation operator is
 \begin{equation}
-  \frac{\partial}{\partial t} J (x,t)
+  \frac{\partial}{\partial t} \hat{\Psi}_\text{H} (x,t)
   =
+  \dots
+\end{equation}
+according to the calculation [[forcebalance:calculation:EoM fermionic field operator|here]].
+Condition:
+\begin{equation}
+  \nu (\vec{r},\vec{r}^\prime)
+  =
+  \nu (\vec{r}^\prime,\vec{r})
+\end{equation}
+=====Particle Number Operator=====
+The EoM of the particle number operator
+\begin{equation}
+  \frac{\partial}{\partial t} \hat{n}_\text{H}=
+  \dots
+\end{equation}
+is of the form of a [[forcebalance:continuity equation]].
+The calculation is done [[forcebalance:calculation:EoM particle number operator|here]].
+=====Current Density Operator=====
+Wtih the help of the EoM of the fermionic field operator, the EoM for the current density operator $J(x,t)$ becomes
+\begin{equation}
+  \frac{\partial}{\partial t} \hat{J}_\text{H} (x,t)
+  =
+  \frac{\partial}{\partial t} \hat{j}_\text{H} (x,t)
+  +
+  \frac{\partial}{\partial t} \hat{j}_\text{d,H} (x,t)
+  =
+  \dots
+  +
   \dots
   \quad .
 \end{equation}
+FIXME Insert Link to para and diramagnetic definitions
 This holds for symmetric correlation potentials
 \begin{equation}
@@ Line 38: / Line 72: @@
   \quad .
 \end{equation}
-The calculation can be found [[forcebalance:calculation:EoM electron current|here]].
+The calculation can be found [[forcebalance:calculation:EoM current density operator|here]].